Large information plus noise random matrix models and consistent subspace estimation in large sensor networks

Introduction

The classical source localization problem consists in estimating vector $\boldsymbol{\theta}=(\theta_{1},\ldots,\theta_{K})^{T}$ from $N$ samples collected in the $M\times N$ matrix $\mathbf{Y}_{N}=({\bf y}_{1},\ldots,{\bf y}_{N})$. This problem was extensively studied in the past (see e.g. and the references therein). The so-called subspace estimator of $\boldsymbol{\theta}=(\theta_{1},\ldots,\theta_{K})^{T}$ is based on the observation that if matrices $\mathbf{A}(\boldsymbol{\theta})$ and $\mathbf{S}_{N}=({\bf s}_{1},\ldots,{\bf s}_{N})$ have both full rank $K$, then the angles $(\theta_{k})_{k=1,\ldots,K}$ are solutions The $K$ angles are the unique solutions under certain assumptions on function $\theta\rightarrow\mathbf{a}(\theta)$ of the equation $\mathbf{a}(\theta)^{*}\boldsymbol{\Pi}_{N}\mathbf{a}(\theta)=0$, where $\boldsymbol{\Pi}_{N}$ represents the orthogonal projection matrix on the kernel of matrix $\mathbf{A}(\boldsymbol{\theta})\mathbf{S}_{N}\mathbf{S}_{N}^{*}\mathbf{A}(\boldsymbol{\theta})^{*}$. The existing subspace methods consist in estimating for each $\theta$ the quadratic form $\eta_{N}(\theta)=\mathbf{a}(\theta)^{*}\boldsymbol{\Pi}_{N}\mathbf{a}(\theta)$ of $\boldsymbol{\Pi}_{N}$ by a certain term $\hat{\eta}_{N}(\theta)$, and then to estimate the $K$ angles as the argument of the $K$ most significant local minima of function $\theta\rightarrow\hat{\eta}_{N}(\theta)$. This approach has been extensively developed when $N\rightarrow+\infty$ and $M$ fixed. In this context, $\eta_{N}(\theta)$ can be estimated consistently for each $\theta$ by $\hat{\eta}_{N}(\theta)=\mathbf{a}(\theta)^{*}\hat{\boldsymbol{\Pi}}_{N}\mathbf{a}(\theta)$ with $\hat{\boldsymbol{\Pi}}_{N}$ the orthogonal projection matrix on the eigenspace associated to the $M-K$ smallest eigenvalues of the empirical covariance matrix $\frac{1}{N}\mathbf{Y}_{N}\mathbf{Y}_{N}^{*}$. It clearly holds that $\sup_{\theta\in[-\pi,\pi]}\left|\hat{\eta}_{N}(\theta)-\eta_{N}(\theta)\right|$ converges torwards 0 almost surely, and this allows to prove that the corresponding estimators $(\hat{\theta}_{k})_{k=1,\ldots,K}$ of the direction of arrivals are consistent.

If however $M$ and $N$ are of the same order of magnitude, a quite common situation if the number of sensors $M$ is large, then the above estimators show poor performances because $\hat{\boldsymbol{\Pi}}_{N}$ is no longer an accurate estimator of $\boldsymbol{\Pi}_{N}$. In order to study this context, Mestre & Lagunas were the first to propose consistent estimators of $\eta_{N}(\theta)$ when $M,N\rightarrow+\infty$ in such a way that $c_{N}=\frac{M}{N}\rightarrow c$, with $c>0$. In Mestre & Lagunas , it is assumed that the source signals $(s_{k,n})_{k=1,\ldots,K}$ are mutually independent complex Gaussian i.i.d. time series with unit variance elements. Under this assumption, $\mathbf{y}_{n}$ can be written as

2 General notations and useful results

We now introduce various notations and results used throughout the paper.

The quantity $C$ will represent a generic positive constant whose main feature is to be deterministic and independent of $M$ and $N$. The value of $C$ may change from one line to another.

We also notice that if $m(z)$ is the Stieltjes transform of positive measure $\mu$, then it holds that

Background on the Information plus Noise model and on the estimator of [23]

Assumption A-1: $0<c_{N}<1\;\mbox{and}\;0<c<1$.

In this section, $\boldsymbol{\Sigma}_{N}$ represents the complex valued $M\times N$ random matrix given by

where $\mathbf{B}_{N}=\frac{\mathbf{A}(\boldsymbol{\theta})\mathbf{S}_{N}}{\sqrt{N}}$ and $\mathbf{W}_{N}=\frac{\mathbf{V}_{N}}{\sqrt{N}}$. Matrices $\mathbf{B}_{N}$ and $\mathbf{W}_{N}$ are assumed to satisfy the following assumptions

Assumption A-2: Matrix $\mathbf{B}_{N}$ is deterministic and satisfies $\sup_{N}\|\mathbf{B}_{N}\|<+\infty$

Assumption A-4: The entries of matrix $\mathbf{W}_{N}$ are i.i.d and follow the complex normal distribution $\mathcal{CN}(0,\frac{\sigma^{2}}{N})$.

We assume moreover that the non zero eigenvalues of $\mathbf{B}_{N}\mathbf{B}_{N}^{*}$ have multiplicities 1 in order to simplify the notations. In the following, we denote by $0=\lambda_{1,N}=\ldots=\lambda_{M-K,N}<\lambda_{M-K+1,N}<\ldots<\lambda_{M,N}$ and $({\bf u}_{k,N})_{k=1,\ldots,M}$ the ordered eigenvalues and associated eigenvectors of $\mathbf{B}_{N}\mathbf{B}_{N}^{*}$. The eigenvalues and the eigenvectors of matrix $\boldsymbol{\Sigma}_{N}\boldsymbol{\Sigma}_{N}^{*}$ are denoted $(\hat{\lambda}_{k,N})_{k=1,\ldots,M}$ and $(\hat{{\bf u}}_{k,N})_{k=1,\ldots,M}$, and $\hat{\mu}_{N}$ represents the empirical eigenvalue distribution of $\boldsymbol{\Sigma}_{N}\boldsymbol{\Sigma}_{N}^{*}$ defined by

As we assume $c_{N}<1$, the joint probability distribution of $(\hat{\lambda}_{k,N})_{k=1,\ldots,M}$ is absolutely continuous (see e.g. James ) and it holds that the $(\hat{\lambda}_{k,N})_{k=1,\ldots,M}$ have multiplicity 1 almost surely. We finally denote by $\mathbf{Q}_{N}(z)$ the resolvent of matrix $\boldsymbol{\Sigma}_{N}\boldsymbol{\Sigma}_{N}^{*}$, i.e. $\mathbf{Q}_{N}(z)=\left(\boldsymbol{\Sigma}_{N}\boldsymbol{\Sigma}_{N}^{*}-z\mathbf{I}_{M}\right)^{-1}$.

It is well-known ([11, Th. 7.4], [9, Th. 1.1]) that it exists a sequence of deterministic probability measures $(\mu_{N})$ such that $\hat{\mu}_{N}-\mu_{N}\to_{N}0$ weakly almost surely. Measure $\mu_{N}$ is characterized by its Stieltjes transform $m_{N}(z)$ which is known to satisfy the equation

with $P_{1}$ and $P_{2}$ two polynomials with positive coefficients, independent of $N$. Then,

Taking into account the previous result, it is shown in that

If we denote by $\mathbf{T}_{N}(z)$ the matrix-valued function defined by

then ${\bf T}_{N}$ coincides with the Stieltjes transform of a positive matrix valued measure $\boldsymbol{\mu}_{N}$ with support $\mathcal{S}_{N}$ such that $\boldsymbol{\mu}_{N}(\mathcal{S}_{N})=\mathbf{I}_{M}$ (see Hachem et al [13, Th. 2.4 & Prop. 2.2]), i.e.

In the remainder of the paper, we will make use of the following result proved in if ${\bf W}_{N}$ is complex Gaussian and in Hachem et al in the non Gaussian case.

Consider two sequences of deterministic vectors $(\mathbf{b}_{N})$, $(\mathbf{d}_{N})$ such that $\sup_{N}\|\mathbf{b}_{N}\|<+\infty$ and $\sup_{N}\|\mathbf{d}_{N}\|<+\infty$. Then, it holds that

In order to present the chacterization of $\mathcal{S}_{N}$, we first introduce the following notations. We denote by $f_{N},\phi_{N}$ and $w_{N}$ the functions defined by

We are now in position to characterize $\mathcal{S}_{N}$.

The function $\phi_{N}$ admits $2Q$ non-negative local extrema counting multiplicities (with $1\leq Q\leq K+1$) whose preimages are denoted $w_{1,N}^{-}<0<w_{1,N}^{+}\leq w_{2,N}^{-}\ldots\leq w_{Q,N}^{-}<w_{Q,N}^{+}$. Define $x_{q,N}^{-}=\phi_{N}(w_{q,N}^{-})$ and $x_{q,N}^{+}=\phi_{N}(w_{q,N}^{+})$ for $q=1\ldots Q$. Then,

and the support $\mathcal{S}_{N}$ of $\mu_{N}$ is given by

Moreover, for $q=1,\ldots,Q$, each interval $]w_{q,N}^{-},w_{q,N}^{+}[$ contains at least an element of the set $\{0,\lambda_{M-K+1,N},\ldots,\lambda_{M,N}\}$ and each eigenvalue of $\mathbf{B}_{N}\mathbf{B}_{N}^{*}$ belongs to one of these intervals.

The second statement of the theorem shows that each eigenvalue of $\mathbf{B}_{N}\mathbf{B}_{N}^{*}$ corresponds to a certain interval of $\mathcal{S}_{N}$. More precisely, an eigenvalue of $\mathbf{B}_{N}\mathbf{B}_{N}^{*}$ will be said to be associated to cluster $[x_{q,N}^{-},x_{q,N}^{+}]$ if it belongs to the interval $(w_{q,N}^{-},w_{q,N}^{+})$. We note that the eigenvalue is necessarily associated to the first cluster $[x_{1,N}^{-},x_{1,N}^{+}]$.

$w_{N}(x_{q,N}^{-})=w_{q,N}^{-}$ and $w_{N}(x_{q,N}^{+})=w_{q,N}^{+}$ for each $1\leq q\leq Q$,

3 Some useful evaluations

In this paragraph, we gather some useful bounds related to certain Stieltjes transforms. We first recall that the inequality

By using the identity, $\mathbf{T}_{N}(z)-\mathbf{T}_{N}(z)^{*}=\mathbf{T}_{N}(z)\left(\mathbf{T}_{N}(z)^{-*}-\mathbf{T}_{N}(z)^{-1}\right)\mathbf{T}_{N}(z)^{*}$, we get after some algebra

Indeed, $\mathbf{T}_{N}(z)$ can be written as $\mathbf{T}_{N}(z)=(1+\sigma^{2}c_{N}m_{N}(z))\left(\mathbf{B}_{N}\mathbf{B}_{N}^{*}-w_{N}(z)\mathbf{I}_{M}\right)^{-1}$. Therefore, $\|\mathbf{T}_{N}(z)\|$ is equal to

Since $\hat{m}_{N}(z)$ is the Stieltjes transform of the distribution $\frac{1}{M}\sum_{k=1}^{M}\delta_{\hat{\lambda}_{k,N}}$, it holds that

where $\mathbf{1}$ denotes vector $\mathbf{1}=(1,1,\ldots,1)^{T}$. We denote $\hat{\omega}_{1,N}\leq\ldots\leq\hat{\omega}_{M,N}$ its eigenvalues. Then we have the following straighforward properties.

The zeros of $z\mapsto 1+\sigma^{2}c_{N}\hat{m}_{N}(z)$ are included in the set $\{\hat{\omega}_{1,N},\ldots,\hat{\omega}_{M,N}\}$.

If the eigenvalues $\hat{\lambda}_{1,N},\ldots,\hat{\lambda}_{M,N}$ of $\boldsymbol{\Sigma}_{N}\boldsymbol{\Sigma}_{N}^{*}$ have multiplicity one, the equation $1+\sigma^{2}c_{N}\hat{m}_{N}(z)=0$ has $M$ multiplicity one solutions which coincide with the $(\hat{\omega}_{k,N})_{k=1,\ldots,M}$. Moreover, $\hat{\lambda}_{1,N}<\hat{\omega}_{1,N}<\ldots<\hat{\lambda}_{M,N}<\hat{\omega}_{M,N}$.

If the eigenvalue $\hat{\lambda}_{k,N}$ has multiplicity $p>1$, i.e. $\hat{\lambda}_{k-1,N}<\hat{\lambda}_{k,N}=\hat{\lambda}_{k+p-1,N}<\hat{\lambda}_{k+p,N}$, then,

and the $\hat{\omega}_{k,N}$ that do not coincide with some eigenvalues of $\boldsymbol{\Sigma}_{N}\boldsymbol{\Sigma}_{N}^{*}$ are zeros of $1+\sigma^{2}c_{N}\hat{m}_{N}(z)$.

Since $c_{N}<1$, we recall that the eigenvalues $(\hat{\lambda}_{k,N})_{k=1,\ldots,M}$ have multiplicity 1 almost surely. However, in subsection 3.2, it will be necessary to define properly the solutions of $1+\sigma^{2}c_{N}\hat{m}_{N}(z)=0$ everywhere. This explains why the case where some of the $(\hat{\lambda}_{k,N})_{k=1,\ldots,M}$ are multiple has to be considered.

Function $z\mapsto\frac{-1}{z(1+\sigma^{2}c_{N}\hat{m}_{N}(z))}$ is the Stieltjes transform of a probability measure whose support coincides with the set of all roots of the equation $z(1+\sigma^{2}c_{N}\hat{m}_{N}(z))=0$, which is included into the set $\{0,\hat{\omega}_{1,N},\ldots,\hat{\omega}_{M,N}\}$. Therefore, it holds that

We recall the two following useful results of and .

for each $N>N_{0}$. Then, with probability one, no eigenvalue of $\boldsymbol{\Sigma}_{N}\boldsymbol{\Sigma}_{N}^{*}$ belongs to $[a,b]$ for $N$ large enough.

It is useful to mention that $\sup_{N}x_{Q_{N},N}^{+}<+\infty$ and that these two theorems are still valid if $b=+\infty$ (see ).

The approach of is valid under the following assumptions.

Assumption A-5: For $N$ large enough, none of the strictly positive eigenvalues of $\mathbf{B}_{N}\mathbf{B}_{N}^{*}$ is associated to the first cluster $[x_{1,N}^{-},x_{1,N}^{+}]$, i.e. $\lambda_{M-K+1,N}>w_{N}(x_{1,N}^{+})$ for $N$ large enough.

Using theorems 3 and 4, we deduce that if $t_{1}^{-},t_{1}^{+},t_{2}^{-},t_{2}^{+}$ are real numbers independent of $N$ satisfying

then, almost surely, for $N$ large enough, it holds that

Assumptions 2.5 and 2.5 thus imply that, almost surely, the smallest $M-K$ eigenvalues of $\boldsymbol{\Sigma}_{N}\boldsymbol{\Sigma}_{N}^{*}$ are separated from the $K$ greatest ones for $N$ large enough in the sense that the 2 sets of eigenvalues are included into 2 disjoint intervals that do not depend on $N$. It is interesting to remark that Assumptions 2.5 and 2.5 are "deterministic conditions" depending only on $\sigma^{2},c_{N}=\frac{M}{N}$,and on the eigenvalues of $\mathbf{B}_{N}\mathbf{B}_{N}^{*}$. If $K$ remains fixed, recent results of Benaych-Rao (see also ) imply that Assumptions A-5 and A-6 hold if and only if $\liminf_{N\rightarrow+\infty}\lambda_{M-K+1,N}>\sigma^{2}\sqrt{c}$. If however $K$ scales with $N$, the derivation of more explicit conditions equivalent to Assumptions 2.5 and 2.5 is still an open problem.

We are now in position to present the consistent estimator of $\eta_{N}$ proposed in . It is based on the observation that

where ${\cal C}$ represents a contour enclosing and not the strictly positive eigenvalues of $\mathbf{B}_{N}\mathbf{B}_{N}^{*}$, and the symbol ${\cal C}^{-}$ means that the contour is oriented clockwise. The estimator of is based on the observation that under Assumptions 2.5 and 2.5, function $w_{N}(z)$ provides such a contour for $N$ large enough. In the following, for $y>0$ and $\epsilon>0$, $\epsilon<\frac{y}{3}$ small enough, we consider the rectangle $\mathcal{R}_{y}$ defined by

and its boundary $\partial\mathcal{R}_{y}$. Then, the properties of function $w_{N}(z)$ (see Proposition 1) imply that for $N$ large enough, the set $w_{N}(\partial\mathcal{R}_{y})$ is a contour enclosing the origin, but not the other eigenvalues of $\mathbf{B}_{N}\mathbf{B}_{N}^{*}$. Therefore, $\boldsymbol{\Pi}_{N}$ can also be written as

because $(\mathbf{B}_{N}\mathbf{B}_{N}^{*}-w_{N}(z)\mathbf{I}_{M})^{-1}=\frac{{\bf T}_{N}(z)}{1+\sigma^{2}c_{N}m_{N}(z)}$. Using (5) and (8) as well as the following lemma

Almost surely, for $N$ large enough, the $M$ solutions $(\hat{\omega}_{k,N})_{k=1,\ldots,M}$ of the equation $1+\sigma^{2}c_{N}m_{N}(z)=0$ satisfy

almost surely for $N$ large enough. In pratice, the above estimator is quite easy to implement because, as the localization of the poles of the integrand in (22) w.r.t. the contour $\partial\mathcal{R}_{y}$ is known (see lemma 2), the contour integral in (22) can be solved, and expressed in closed form in terms of the $(\hat{{\bf u}}_{k,N},\hat{\lambda}_{k,N},\hat{\omega}_{k,N})_{k=1,\ldots,M}$.

From now on, we assume that vector $\mathbf{a}(\theta)$ is given by (2) and that assumptions 2.5 and 2.5 hold. We consider $t_{1}^{-}$, $t_{1}^{+}$, $t_{2}^{-}$ and $t_{2}^{+}$ satisfying (18) as well a rectangle $\mathcal{R}_{y}$ defined by (20). We prove here the following result.

Assume assumptions A-1 to A-6 hold. Then, we have

In the following, we denote by $\mathcal{T}_{\epsilon}$ the set

We first establish in Sections 3.1 and 3.2 that the events $\mathcal{E}_{1,N}$ and $\mathcal{E}_{2,N}$ defined by

and $\phi(\lambda)\in(0,1)$ elsewhere, and define the random variable

The purpose of this section is to prove the following technical result.

and $\psi_{0}(\lambda)\in(0,1)$ elsewhere. From this definition, we clearly have

we finally obtain that (29) holds for $l=1$.

The first term of the r.h.s of (31) can be upperbounded as follows

Using Hölder’s inequality, we get immediately that

Plugging the previous estimates into (32), we get

In this section, we will prove the following result.

We follow the same approach than in Section 3.1 and first prove that the $(\hat{\omega}_{k,N})_{k=1,\ldots,M}$ satisfy a property similar to (7). For this, we study the behaviour of the Stieltjes transform $\hat{n}_{N}(z)$ of the distribution $\frac{1}{M}\sum_{k=1}^{M}\delta_{\hat{\omega}_{k,N}}$ defined by

and use Lemma 1 as well as the inverse Stieltjes transform formula (3). Our starting point is the following result showing that the empirical eigenvalue distribution of $\hat{\boldsymbol{\Omega}}_{N}$ is very similar to the distribution of the eigenvalues of $\boldsymbol{\Sigma}_{N}\boldsymbol{\Sigma}_{N}^{*}$. The following auxiliary result will be useful.

Assume assumptions A-1 to A-6 hold. It holds that

Proof: The proof is given in Appendix 5.1. $\square$

We now prove the fundamental following result.

Proof: Using that $\hat{\boldsymbol{\Omega}}_{N}$ is a rank 1 perturbation of $\hat{\boldsymbol{\Lambda}}_{N}$, we obtain immediately that

In order to study the expectation of this expression, we use (11) and (15). Moreover, (6) and a straightforward application of the Poincaré inequality to $\hat{m}_{N}(z)$ considered for $z$ fixed as a function of the entries of $\mathbf{W}_{N}$ leads immediately to

This immediately implies (35). Now define the function $h_{N}(z)$ by

When $y\rightarrow 0$, this converges towards $\log(1+\sigma^{2}c_{N}m_{N}(x_{q,N}^{+}))-\log(1+\sigma^{2}c_{N}m_{N}(x_{q,N}^{-}))$, a real quantity because $x_{q,N}^{-}$ and $x_{q,N}^{+}$ belong to $\partial\mathcal{S}_{N}$. This shows that $\kappa_{N}([x_{q,N}^{-},x_{q,N}^{+}])=0$. Consequently,

where $\hat{\boldsymbol{\Pi}}_{l,N}$ represents the orthogonal projection matrix on the 1–dimensionnal eigenspace associated to the eigenvalue $\hat{\lambda}_{l,N}$ of $\boldsymbol{\Sigma}_{N}\boldsymbol{\Sigma}_{N}^{*}$.

a property also established in the appendix. We now prove the following result.

Assume assumptions A-1 to A-6 hold. It holds that

Indeed, if $(\hat{{\bf v}}_{k,N})_{k=1,\ldots,M}$ represent the eigenvectors of $\hat{\boldsymbol{\Omega}}$, then

As $\sum_{k=1}^{M}|{\bf e}_{l}^{T}\hat{{\bf v}}_{k,N}|^{2}=1$, Jensen’s inequality yields to (41). Therefore, it holds that

As $\sup_{N}\|\mathbf{B}_{N}\mathbf{B}_{N}^{*}\|<+\infty$, we get using lemma 7 that

We remark that $\|\boldsymbol{\Sigma}_{N}\boldsymbol{\Sigma}_{N}^{*}\|<t_{2}^{+}+\epsilon$ on the set $\mathcal{E}_{1,N}^{c}$,and write the righthandside of (42) as

for each integer $p$. This completes the proof of lemma 9. $\square$

Assume that (38) holds until integer $l-1$. We write as previously that

The Cauchy-Schwarz inequality leads immediately to

As for the second term of the r.h.s. of (43), we use Poincaré inequality and Hölder’s inequality to obtain

3 End of the proof of theorem 5

We now complete the proof of Theorem 5 when function $\theta\rightarrow\mathbf{a}(\theta)$ is given by

for $\theta\in[-\pi,\pi]$. We recall that $\mathcal{E}_{N}$ is defined by

Assume assumptions A-1 to A-6 hold. For each $N$, it holds that

and remark that for each $\theta\in[-\pi,\pi]$ and for each $N$, there exists $\theta_{N}\in\vartheta_{N}$ such that $|\theta-\theta_{N}|\leq\frac{2\pi}{N^{2}}$. For each $\theta\in[-\pi,\pi]$, it holds that

It is easy to check that the third term of the r.h.s. of (45) satisfies

In order to evaluate the behaviour of the supremum over $\theta$ of the first term of the r.h.s. of (45), we prove that for each $\alpha>0$,

for some constant term $C$. Inequality (46) thus implies that

for each integer $l$. Borel-Cantelli’s lemma eventually implies that

We finally study the supremum of the second term of (45). We denote by $\nu_{k,N}$ the elements of $\vartheta_{N}$. Let $\alpha>0$, then

In order to complete the proof of Theorem 5, we establish the following proposition.

where the constant $C$ does not depend on the sequence $(\mathbf{a}_{N})$.

Proof: In order to shorten the notations, we denote by $\hat{g}_{N}(z)$ and $g_{N}(z)$ the functions defined by

If we denote by $\mathcal{A}_{1,N}$ and $\mathcal{A}_{2,N}$ the events defined by

then $[\mathbf{D}_{1}]_{i,j,N}=0$ on $\mathcal{A}_{1,N}^{c}$ and $[\mathbf{D}_{2}]_{i,j,N}=0$ on $\mathcal{A}_{2,N}^{c}$.

We now establish (49) by induction on $l$, and first consider the case $l=1$. We write the second moment of $(\hat{g}_{N}(z)-g_{N}(z))\chi_{N}^{2}$ as

As $\chi_{N}\neq 0$ implies that $\hat{\lambda}_{M,N}=\|\boldsymbol{\Sigma}_{N}\boldsymbol{\Sigma}_{N}^{*}\|\leq t_{2}^{+}+2\epsilon$, Lemma 10 implies that

The same conclusions hold when the derivatives w.r.t. variables $\overline{W}_{i,j,N}$ are considered. This shows that the first term of the r.h.s. of (52) is a $\mathcal{O}\left(\frac{1}{N}\right)$ term. We now evaluate the behaviour of the second term of the r.h.s. of (52), and establish that

for each integer $p$. We express $\frac{\partial\chi_{N}^{2}}{\partial W_{i,j,N}}$ as $2\chi_{N}\frac{\partial\chi_{N}}{\partial W_{i,j,N}}$. Lemma 10 implies that $\sup_{z\in\partial\mathcal{R}_{y}}\chi_{N}^{2}|\hat{g}_{N}(z)-g_{N}(z)|^{2}<C$. Therefore, it is sufficient to check that

for each integer $p$. $\frac{\partial\chi_{N}}{\partial W_{i,j,N}}$ can be written as

for each integer $p$. Using similar calculations and Proposition 3, we obtain that

for each integer $p$. This completes the proof of (55) and establishes that

Assume assumptions A-1 to A-6 hold. It holds that

We express $(\hat{g}_{N}(z)-g_{N}(z))\chi_{N}^{2}$ as $\beta_{1,N}(z)+\beta_{2,N}(z)$ where

Using Lemma 10, (59) for $\beta_{1,N}$ will be established if we show that

This completes the proof of (59) for $\beta_{1,N}$. In order to show (59) for $\beta_{2,N}$, we first remark that by Lemma 10, $|\mathbf{a}_{N}^{*}\mathbf{T}_{N}(z)\mathbf{a}_{N}|$ is uniformly bounded on $\partial\mathcal{R}_{y}$, and write that

The Poincaré inequality and Lemma 12 imply that

for some deterministic constant $C$ (see Lemma 10). This completes the proof of (49) for $l=1$.

We now assume that (49) holds until integer $l-1$ and write that

The Cauchy-Schwarz inequality implies that

Consistency of the angular estimates

For $k=1,\ldots,K$, with probability one,

In order to establish the proposition, we follow a classical approach initiated by Hannan to study sinusoid frequency estimates. For this, we first recall the following useful lemma.

Appendix

We first give the following useful technical result. Its proof, based on Poincaré’s inequality, is elementary and therefore omitted.

Moreover, the same results still hold when $\mathbf{Q}_{N}(z)$ is replaced by $\mathbf{Q}_{N}(z)^{2}$.

We are now in position to establish Lemma 4. We have to establish that

We first notice that (62) is equivalent to

In order to prove (63), we first show that

where $\boldsymbol{\Delta}_{N}(z)$ is given by $\boldsymbol{\Delta}_{N}(z)=\boldsymbol{\Delta}_{1,N}(z)+\boldsymbol{\Delta}_{2,N}(z)+\boldsymbol{\Delta}_{3,N}(z)$ with

In order to complete the proof of the lemma, we establish that

Proof: We first observe that (66) and (67) imply that

We first need to establish the following useful Lemma.

Therefore, it holds that $\max(|f(x_{n})-f(y_{n})|,|f(z_{n})-f(x)|)<d_{n}^{2}$. Writing $f(x_{n})-f(x)=f(x_{n})-f(y_{n})+f(y_{n})-f(z_{n})+f(z_{n})-f(x)$, we obtain that $f(x_{n})-f(x)=f(y_{n})-f(z_{n})+o(d_{n})$. By differentiability of $f$ on ${\mathcal{O}}$ and continuity of $f^{\prime}$ at $x$,

Using Lemma 4.6 in Haagerup-Thorbjornsen , we obtain

Proof: We start by observing that for any integers $m_{1},m_{2},\ldots,m_{t}$, matrix $\mathbf{A}=\hat{\boldsymbol{\Lambda}}_{N}^{m_{1}}{\bf 11}^{T}\hat{\boldsymbol{\Lambda}}_{N}^{m_{2}}\cdots{\bf 11}^{T}\hat{\boldsymbol{\Lambda}}_{N}^{m_{t}}$ writes

Using Theorem 6 and for $p\geq 2$ the inequality,

4 Proof of Lemma 11: differentiability of the regularization factor

and it remains to apply the composition formula for differentials to obtain (50).

hence the derivative (50) is zero on $\mathcal{A}_{1,N}^{c}$.

if $\hat{\lambda}_{k,N}=\hat{\lambda}_{l,N}$. Indeed, given $\varepsilon>0$, let $\phi_{\varepsilon}(x)=\phi(x)+\varepsilon$. Since $\phi_{\varepsilon}(\hat{\boldsymbol{\Omega}}_{N})>0$,

5 Proof of lemma 12: various estimates

Thus, (80) follows immediately from (13). $\square$

Lemma 18 immediately implies that the following uniform bounds hold.

Proof: We first recall that inequality (10) holds. Therefore, the uniform convergence result (79) implies that

for $N$ large enough. This establishes (82) that holds for $N$ large enough. In order to prove (83), we express $\mathbf{R}_{r,N}(z)$ as

and use (78) and (80). The proof of (84) is similar, and is based on the identity

and that $|w_{r,N}(z)|$ and $|w_{N}(z)|$ are uniformly bounded from below by (13) and (80) (recall that is one of the eigenvalues of $\mathbf{B}_{N}\mathbf{B}_{N}^{*}$). $\square$

with $\sup_{z\in\partial\mathcal{R}_{y}}|\epsilon_{i,N}(z)|\leq C<\infty$.

As for the use of the Poincaré inequality, we have:

Moreover, the same kind of uniform bounds still hold when $\mathbf{Q}_{N}(z)$ is replaced by $\mathbf{Q}_{N}(z)^{2}$.

The proofs of these results are based on elementary arguments, and are thus omitted. Following the calculations of and , we obtain that

After some calculations using Lemmas 19, 20, 21, we eventually obtain that

for all large $N$. In order to prove (56) and (57), it remains to handle the terms involving the difference $\mathbf{R}_{r,N}(z)-\mathbf{T}_{N}(z)$. We show in the following that

for all large $N$. We start as usual with the identity $\mathbf{R}_{r,N}(z)-\mathbf{T}_{N}(z)=\mathbf{R}_{r,N}(z)\left(\mathbf{T}_{N}(z)^{-1}-\mathbf{R}_{r,N}(z)^{-1}\right)\mathbf{T}_{N}(z)$, to get

There exists a constant $C>0$ independent of $N$ such that

Proof: It is shown in and that $\underline{\Delta}_{N}(z)$ is the determinant of the following $2\times 2$ linear system

We deduce from this that $\inf_{z\in\partial\mathcal{R}_{y}}\left|\Delta_{N}(z)\right|\geq C>0$ for all large $N$. Therefore, we can invert the system (95) and obtain

for all large $N$. This establishes (57) and completes the proof of (56).

References